Getting really fine details without taxing too much VRAM?
The 1px adaptive displacement is the more accurate one. The reason you see it as worse is because it exposes your displacement map for what it is: an 8bit file (and one with an odd value range at that).

Your standard 8bit integer images store values as integers (whole numbers) from 0-255. That means an 8bit integer displacement map has only 256 possible height levels. Worse, your map is very low contrast. The 'blank' value is 128 (I think you might have another misconception here, see below). The lowest value I could find in a quick check is 108, meaning your entire tire track has only 20 different heights in it. Obviously, you're not going to get smooth gradients over only 20 possible heights!

Unfortunately, painting in Photoshop is rasterized immediately, so you'll have to repaint in a 16 or 32bit file to fix this. Even just using 16bit gives you 2^8 more height possibilities than 8bit. And remember, you were only using about 1/10th of the 8bit range. So make a solid white 16bit file, and paint some tire tracks with a heavy hand, the deepest point in the track should be close to black.

Above I commented on your use of 128 as a "blank' value and recommended using full white on the repaint. The reason for this is how Cycles interprets the displacement height. Cycles' displacement system is designed around floating point inputs, and simply takes the value given as literal distance. Meaning the "blank" value is black, and negative numbers need to be used to displace downwards. Simply Photoshop isn't really conducive to painting negative values, it's easiest to paint in a 0-1 range then just subtract 1: 

As far as memory savings go, this answer of mine might be useful as well: Render fails at tesselation with out of memory error